Overview

In this project I will investigate the exponential distribution in R and compare it with the Central Limit Theorem(CLT). CLT tells us that sample mean distribution is a normal distribution when sample size is large enough (for example >30). Expected value of sample mean distribution is used to estimate population mean. Sample mean variance is popolation variance divied by sample size.

Simulations

We’ll generate 1000 exponential distribution samples with each sample size of 40. We’ll set lambda to 0.2 as required.

n <- 40
lambda <- 0.2
exp.sample.means <- NULL
for (i in 1:1000) {
    exp.sample <- rexp(n, lambda)
    mean(exp.sample)
    exp.sample.means <- c(exp.sample.means, mean(exp.sample))
}
dat <- data.frame(x=exp.sample.means)

Sample Mean versus Theoretical Mean

Mean of sample means:

sample.mean <- mean(exp.sample.means)
sample.mean
## [1] 5.010925

Exponential distribution has a theoretical mean of 1/lambda.

theoretical.mean <- 1 / lambda
theoretical.mean
## [1] 5

So we can see what CLT tells, the sample mean 5.0109252 can be used to estimate theoretical mean 5.

We can plot sample mean distribution as following. X axis is values of sample means and Y axis is the counts of each value(Frequency). The distribution is centered at sample mean 5.0109252 which is highlighted with the red line. The theoretical mean 5 is highlighted with the green line.

library(ggplot2)
g <- ggplot(dat, aes(x = x)) + 
     geom_histogram(fill="blue", col="black", binwidth = 0.2) +
     ggtitle("Sample Mean Distribution of Exponential distribution") +
     xlab("Sample Mean") +
     ylab("Frequency")
g <- g + geom_vline(xintercept = sample.mean,  col = "red")
g <- g + geom_vline(xintercept = theoretical.mean, col = "green")
g

Sample Variance versus Theoretical Variance

Sample variance can be calculated as following:

sample.var <- var(exp.sample.means)
sample.var
## [1] 0.6095526

Theoretical standard deviation of a exponential population is 1/lambda. so Theoretical variance can be calculated. As CLT tells us, sample variance is population variance divided by sample size. We can calculate theoretical sample variance as following:

theoretical.sd <- 1 / lambda
theoretical.var.population <- theoretical.sd ^ 2

theoretical.var.sample <- theoretical.var.population / n
theoretical.var.sample
## [1] 0.625

As we can see, sample variance 0.6095526 is very close to theoretical variance 0.625

Distribution

A Q–Q plot is used to compare the shapes of distributions - commonly used to compare a data set to a theoretical model. If the two distributions being compared are similar, the points in the Q–Q plot will approximately lie on the line y = x. We’ll use qqnorm and qqline to plot our sample mean distribution compared with normal distribution. We can see they are quite similar in terms of quantiles against each other. So we can say our sample mean distribution is very close to normal distribution.

qqnorm(exp.sample.means) 
qqline(exp.sample.means)

Refrences

  1. https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot